Find $\dfrac{d}{dx}[2\log_5(x)]$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{2}{x}$ (Choice B) B $\dfrac{2}{5x}$ (Choice C) C $\dfrac{2}{\ln(5)x}$ (Choice D) D $\dfrac{2}{x\log_5(x)}$
Solution: The expression to differentiate includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative as shown below. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}[2\log_5(x)] \\\\ &=2\dfrac{d}{dx}[\log_5(x)] \\\\ &=2\cdot\dfrac{1}{\ln(5)x} \\\\ &=\dfrac{2}{\ln(5)x} \end{aligned}$ In conclusion, $\dfrac{d}{dx}[2\log_5(x)]=\dfrac{2}{\ln(5)x}$.